$\frac{1}{2}\log x^2+\frac{1}{3}\log x^3=\frac{1}{2}\log x^{10}-\frac{1}{3}\log x^9$  

 

$L=\frac{1}{2}\log x^2+\frac{1}{3}\log x^3=\log {\left(x^2\right)}^{\frac{1}{2}}+\log {\left(x^3\right)}^{\frac{1}{3}}=\log x^{2\cdot \frac{1}{2}}+\log x^{3\cdot \frac{1}{3}}=\log x+\log x=\log \left(x\cdot x\right)=\log x^2=2\log x$  

 

   

$P=\frac{1}{2}\log x^{10}-\frac{1}{3}\log x^9=\log {\left(x^{10}\right)}^{\frac{1}{2}}-\log {\left(x^9\right)}^{\frac{1}{3}}=\log x^{10\cdot \frac{1}{2}}-\log x^{9\cdot \frac{1}{3}}=\log x^5-\log x^3=\frac{\log \left(x^5\right)}{x^3}=\log x^2=2\log x$  

 

$L=P$